Braid of braids groups and evolution algebras.

Abstract

The purpose of this Thesis is to develop the theory of braid of braids and twin of twin groups. We investigate some of their fundamental structural aspects in detail. We will attach particular importance to the reduced presentations of many of these groups. The term braid of braids identifies classes of algebraic groups of braids that each as include a particular case, in a certain sense, a class of braids representing one of the classical braid groups. The term indicates that each strand of this type of group that uses it is generally made up of several strands that we will call elementary and ”borrowed” from classic braid groups. E. Artin introduced in 1925 the braid groups Bn as a tool for working with classical knots and links. In recent years, there has been an increase in the number of publications related to this type of group. The manuscript consists of the following chapters: 1. Introduction 2. Preliminaries 3. Braid of braid groups 4. Pseudosymmetric braid of braid 5. Singular pseudosymmetric braid of braid 6. Virtual Singular Pseudosymmetric braid of braids 7. Virtual Singular Pseudosymmetric twin of twins 8. Unrestricted Virtual Singular Pseudosymmetric twin of twins 9. Evolution Algebra. We discussed the presentations of these groups, the various extensions and homomorphisms between them. In particular, we have studied four reduced presentations of this type of structure. The last part of the thesis, Chapter 9, deals with evolution algebras. An evolving field of research. We have shown that the category of evolution algebras over the field k is finitely universal.